# Matrix factorization method

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matrix forward-backward substitution method

A method for solving finite-difference systems that approximate boundary value problems for systems of ordinary differential equations in one-dimensional problems, and for elliptic equations in two-dimensional problems.

The solution of the three-point difference scheme

where is an unknown grid vector, is the right-hand side vector and are given square matrices, under the boundary conditions

is sought for, as in the scalar case, in the form

 (*)

The coefficients (the matrix and the vector ) are determined from the recurrence relations ( "forward substitution" )

while and are given by the left boundary condition:

The are calculated by formula (*) ( "backward substitution" ), and

There is stability of this method to rounding errors under the conditions

which implies that , (see [1]). A different form of the stability conditions is also available (see [2], [3]). The matrix factorization method is applied also to two-point difference schemes (see [3]). A variant in which inversion of matrices is replaced by orthogonalization is also used (see [4]).

#### References

 [1] A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian) [2] V.V. Ogneva, "The "sweep" method for the solution of difference equations" USSR Comp. Math. Math. Phys. , 7 : 4 (1967) pp. 113–126 Zh. Vychisl. Mat. i Mat. Fiz. , 7 : 4 (1967) pp. 803–812 [3] A.A. Samarskii, E.S. Nikolaev, "Numerical methods for grid equations" , 1–2 , Birkhäuser (1989) (Translated from Russian) [4] S.K. Godunov, "A method of orthogonal successive substitution for the solution of systems of difference equations" USSR Comp. Math. Math. Phys. , 2 : 6 (1962) pp. 1151–1165 Zh. Vychisl. Mat. i Mat. Fiz. , 2 : 6 (1962) pp. 972–982 [5] E.L. Wachspress, "Iterative solution of elliptic systems and applications to the neutron diffusion equations of reactor physics" , Prentice-Hall (1966)